Injaz, who is said to be mentally ill, attacked the embassy with a knife, a toy gun and a can of petrol, sparking a major security alert in Tel Aviv.
An Israeli Arab lawyer who spoke to the man by phone to try to calm him down said the Palestinian had held hostage the Turkish consul-general and his wife for some two hours.
They managed to escape after Turkish security officers shot and wounded the Palestinian.
Israeli media said it appeared the man had managed to enter the embassy through an open window on the ground floor.
Al Jazeera's Nisreen el-Shamayleh, reporting from Tel Aviv, said it appeared the Turkish embassy had taken the lead in dealing with the attack.
Israeli television Channel 2 played a voice recording of Injas, in which he said he would destroy the embassy unless he was given asylum in Turkey. He said he had weapons, explosives and gasoline.
An Israeli foreign ministry spokesman said that it was not clear what would happen to the man.
An Israeli Foreign Ministry official said the man had been released from an Israeli jail two weeks ago after completing a four-year sentence for breaking into the British embassy in 2006 to demand asylum.
Israeli authorities said that Injaz had attacked the British embassy in Israel four years ago, in a similar bid to claim asylum by force.
At the time of the first attack, police said Injaz was an informer and a criminal with a record of property and drug offenses who had run into financial and legal troubles.
He told an Israeli TV station in 2006 that he feared he would be killed by Palestinian armed groups if he returned to the West Bank.
The attack comes at a time of strained ties between Israel and Turkey, which recalled its ambassador after Israeli troops killed nine Turkish humanitarian activists on board an aid ship bound for Gaza in May.